Cash-flow Risk, Discount Risk, and the Value Premium

Transcription

1 Cash-flow Risk, Discoun Risk, and he Value Premium Tano Sanos Columbia Universiy and NBER Piero Veronesi Universiy of Chicago, CEPR and NBER June 3, 2005 Absrac We propose a general equilibrium model wih muliple asses ha explicily ies he ime series properies of he marke porfolio wih he cross-secional predicabiliy of reurns on price sored porfolios, he value premium. This link provides igh predicions abou he cash-flow characerisics of value and growh socks, such as heir fundamenals cash-flow risk and heir expeced dividend growh. We show ha (a subsanial crosssecional difference in fundamenals cash-flow risk is necessary o generae a plausible value premium; (b he ime variaion he aggregae expeced reurn induces flucuaions in he size of he value premium and ha, in paricular, he value premium is larger in bad imes; (c he uncondiional CAPM fails, and hus a value premium puzzle arises, because of general equilibrium resricions on he marke porfolio; and (d an HML facor lines up reurns as i capures aggregae differences in cash-flow risk in he economy. Our model also sheds ligh on he empirical performance of condiional asse pricing models ha have been recenly proposed o address he value premium. VERY PRELIMINARY AND INCOMPLETE. PLEASE DO NOT CIRCULATE WITHOUT PERMISSION.

2 I. INTRODUCTION Hisorically, socks wih high book-o-marke raios, value socks, have yielded higher average reurns han socks wih low book-o-marke raios, growh socks. The CAPM s major failure is is inabiliy o price book-o-marke sored porfolios. As a resul of his failure, a large collecion of explanaions have been proposed o address his value premium puzzle, from he behavioral o he raional ones ha argue ha he value premium is simply a compensaion for he larger risk of value socks. 1 These explanaions hough are surprisingly deached from he voluminous lieraure ha focuses on he properies of he aggregae marke porfolio, such as he large equiy premium and he high volailiy and predicabiliy of aggregae reurns. In his paper we argue ha he ime series behavior of he marke porfolio imposes general equilibrium resricions on he behavior of he cross-secion of average reurns of price sored porfolios. These resricions are imporan as hey provide igh implicaions abou he cashflow characerisics of value and growh socks as well as abou he variaion over ime of he value premium iself. Our predicions are broadly consisen wih empirical evidence. More specifically, we adop a represenaive agen economy where preferences are of he exernal habi persisence ype inroduced by Campbell and Cochrane (1999. These preferences are successful in generaing plausible quaniaive implicaions for he marke porfolio hrough he ime variaion of he marke price of consumpion risk. We follow Menzly, Sanos and Veronesi (2004, MSV henceforh, and embed hese preferences in a general equilibrium seing wih muliple risky asses. 2 These asses have ime varying expeced dividend growh and differ from each oher in heir cash-flow risk, ha is, in he covariance of heir cash-flow wih he aggregae economy. By generalizing some resuls of MSV, we show ha (a subsanial cross-secional differences in cash-flow risk are necessary o generae a plausible value premium; (b he ime variaion in he marke price of consumpion risk ineracs wih he cross-secional differences in cash-flow risk o induce flucuaions in he value premium and ha, in paricular, he value premium is high whenever he marke premium is also high; (c because of general equilibrium resricions on he oal wealh porfolio, he uncondiional 1 For he value premium see Rosenberg, Reid, and Lansein (1985 and Fama and French (1992. For behavioral explanaions see for example Rosenberg, Reid, and Lansein (1985, DeBond and Thaler (1987 and Lakonishok, Shleifer, and Vishny (1994. For he raional ones see Fama and French (1993, Leau and Ludvigson (2001, Gomes, Kogan and Zhang (2003 among ohers. 2 To he bes of our knowledge he firs fully fledged general equilibrium model of he cross secion of sock reurns is Gomes, Kogan, and Zhang (2003 who build on he parial equilibrium model of Berk, Green and Naik (1999. See also Zhang (

3 CAPM fails and hus a value premium puzzle obains; and (d an HML facor, as in Fama and French (1993, lines up reurns as i capures aggregae differences in cash-flow risk in he economy. In addiion, he model also sheds ligh on he performance of he recenly proposed condiional CAPM models. In order o undersand he inuiion of our resuls, consider firs he case where all asses have idenical cash-flow risk. 3 In his case, we show ha wheher an asse has a high or low premium depends on he asse s duraion, by which we mean wheher i has low or high expeced cash-flow growh. Asses wih high expeced cash-flow growh are relaively more sensiive o shocks in he aggregae discoun han oherwise idenical asses wih low expeced cash-flow growh. 4 When only discoun effecs are presen hen, he risk-reurn rade-off is only deermined by he iming of cash-flows, ha is by he asse s duraion. Can discoun effecs generae he value premium? No. Asses wih srong expeced cash-flow growh have high price-dividend raios and, as jus menioned, a high sensiiviy o changes in he aggregae discoun, and, as a consequence, command a higher premium. Thus, a counerfacual posiive relaion obains beween price-dividend raios and average excess reurns. I follows ha if he value premium is o obain, cross-secional differences in cash-flow risk have o be sufficienly srong o undo discoun effecs. Indeed, consider now he case of a low duraion asse whose cash-flow growh is srongly posiively correlaed wih he growh rae of he aggregae economy. In his case, and due o is low expeced dividend growh, he oal value of his asse is mainly deermined by he curren level of cash-flows, raher han by hose in he fuure. The price of he asse is hen mosly driven by cash-flow shocks and he fundamenal risk embedded in hese cashflows drives also he risk of he asse. Thus, when cash-flows display subsanial fundamenal risk, he asse s premium is higher when he duraion is lower. Can hese cash-flow effecs generae he value premium? Yes. Asses wih high cash-flow risk and low duraion have low price-dividend raios. This is due o boh he fac ha hey are risky, and hus prices have o be low o compensae agens for he risk hey ake, and because hey have low expeced dividend growh. Thus, poenially, he value premium can now arise, and wheher i does or no depends on how he ension beween discoun effecs (high risk when he asse has 3 In a general equilibrium seing, all asses have idenical cash-flow risk only if heir covariance of dividends wih consumpion equals he variance of consumpion. Tha is, all asses have a cash-flow risk ha equals he one of he aggregae endowmen iself. 4 The parallel wih he sandard inuiion in fixed income is helpful here: he price of a zero discoun bond of longer mauriy is more sensiive o shocks in he aggregae discoun han one wih shorer mauriy. 2

4 a high duraion and cash-flow effecs (high risk when he asse has low duraion resolves quaniaively. In addiion o an uncondiional value premium, our model provides predicions for is dynamics over ime. We show ha discoun risk effecs inerac wih he cross-secional dispersion in cash-flow risk o induce ime series variaion in he relaive risk of value versus growh socks over he business cycle as measured by shocks o aggregae consumpion. In paricular, consisenly wih he daa, value socks are paricularly risky during bad imes. This is naural: Agens demand a relaively higher compensaion for holding asses wih cash-flows ha covary posiively wih consumpion growh when faced wih adverse shocks. Thus cashflow effecs have a condiional effec as well. To pu i differenly, cross-secional dispersion in cash-flow risk resuls no only on cross-secional dispersion in uncondiional average excess reurns, bu also on flucuaions of he value premium ha reinforce he uncondiional effecs. To evaluae he model s abiliy o yield quaniaively plausible implicaions we simulae an economy wih 200 asses over 10,000 years of quarerly daa. Throughou we mimic he procedure employed in he lieraure of soring asses ino decile porfolios formed on he basis of price-dividend raios. We calibrae he discoun parameers o mach he ime series properies of he aggregae marke porfolio and hus generae quaniaively relevan discoun effecs. We hen calibrae he cash-flow effecs o obain properies of he cross-secion of sock reurns ha mach well hose in he daa. In order o do so, we assume ha he crosssecional dispersion in cash-flow risk is symmerically disribued around zero. In our model, we find ha in order o obain he value premium some of he underlying asses need o have cash-flow growh processes ha are, in absolue value, srongly correlaed wih consumpion growh. 5 Tha is, he srong discoun effecs ha are needed o generae he sandard baery of momens for he aggregae porfolio imply ha also very srong cash-flow effecs are needed o generae he sandard baery of momens for he cross-secion of price sored porfolios. Moreover, we show ha he preferred parameerizaion also generaes he flucuaions in he value premium observed in he daa. We hen proceed o inerpre he success (and failure of exising asse pricing models, such as he CAPM, he condiional CAPM, and he Fama and French (1993 model in he conex of our framework. Firs, in our seup, he CAPM does no hold eiher condiionally or uncondiionally. The reason is ha general equilibrium resricions induce a mild predicabiliy 5 In a general equilibrium seing, if some socks are srongly correlaed wih consumpion growh, ohers mus have a srong negaive correlaion wih consumpion growh. 3

5 in expeced consumpion growh and his breaks he perfec correlaion beween he sochasic discoun facor and he reurn of he marke porfolio. We show ha he CAPM performs poorly in our simulaions bu ha he condiional CAPM, which is a misspecified asse pricing model in our seup, performs much beer. The reason is well known: The condiional asse pricing model capures he increase in he relaive riskiness of value socks in bad imes, ha is, whenever he marke premium is high. Bu his effec can only arise if value socks are also he asses ha have high cash-flow risk uncondiionally, as our simulaions show. This provides a raionale as o why he wo differen srands of raional explanaions ha have been recenly proposed o address he value premium, condiional asse pricing models such as Leau and Ludvigson (2001 and models based on cash-flow risk, such as Parker and Julliard (2005 and Bansal, Dimar, and Lundblad (2005, have boh been shown o be relaively successful: They are boh sides of he same coin in he presence of discoun effecs. We hen urn our aenion o he facor model of Fama and French (1993. In paricular we consruc an HML facor which capures he cross-secional dispersion in cash-flow risk. Asses wih a high cash-flow risk have, mechanically, a high loading on HML. The premium on HML varies over ime, as already menioned, so he Fama and French model performs well because i capures he sources of uncondiional cross-secional variaion in average excess reurns hrough he loadings on HML as well as he sources of condiional variaion hrough he flucuaions in he premium on HML. Finally, when we run in simulaed daa he ime series regressions of he reurns of he price sored porfolios on he reurns on HML, we obain loadings ha are very similar o heir empirical counerpars. The paper proceeds as follows: Nex secion inroduces he model. Secion III conains he model resuls for prices and expeced reurns. In paricular, i discusses he source of he value premium in our seing. Secion IV conains a simulaion of our model and comparison o daa and evaluaes he model s abiliy o mach basic momens of he reurns daa, boh in he ime series and he cross secion. In paricular we invesigae he ime series behavior of he value premium as a funcion of he dividend yield of he marke porfolio. Secion V evaluaes and inerpres he exising asse pricing models in ligh of our findings. Secion VI quanifies he magniudes of he cash-flow risk effecs ha are needed o generae he value premium. Secion VII offers some exensions and addiional discussions and VIII concludes. All proofs are in Appendix. 4

6 II. THE MODEL II.A Preferences There is a represenaive invesor who maximizes [ ] E u (C, X, d, (1 where he insananeous uiliy funcion is give by 0 u (C, X, = { e ρ (C X 1 γ 1 γ if γ > 1 e ρ log (C X if γ = 1 (2 In (2, he variable X denoes an exernal habi level and ρ denoes he subjecive discoun rae. 6 The exac specificaion of he exernal habi X is described below. II.B Cash-flows We consider an endowmen economy wih n financial asses. Each asse has an insananeous dividend sream denoed by D, i for i = 1,.., n. The aggregae endowmen available for consumpion a any ime is hen equal o he sum of dividends. 7 The consumpion good is immediaely perishable and non-sorable, which yields he equilibrium resricion C = D i (3 i=1 Equaion (3 implies ha specific assumpions made on he dividend processes immediaely ranslae ino paricular dynamics for aggregae consumpion. Unforunaely, even when one assumes relaively simple processes for D i he resuling process for aggregae consumpion is difficul o work wih and resricive assumpions need o be made for racabiliy. 8 In order 6 On habi persisence and asse pricing see Sundaresan (1989, Consaninides (1990, Abel (1990, Ferson and Consaninides (1991, Deemple and Zapaero (1991, Daniel and Marshall (1997, Campbell and Cochrane (1999, Li (2001, and Wacher (2000. These papers hough only deal wih he ime series properies of he marke porfolio and have no implicaions for he risk and reurn properies of individual securiies. 7 For consisency wih he daa, we should consider also oher forms of income such as labor income. Alhough doing so does no presen any addiional echnical challenge, we assume noneheless ha oal consumpion equals dividends as he resuls and he general equilibrium resricions are easier o see in his case. In Sanos and Veronesi (2005 we provide a horough discussion of he role of labor income in a framework similar o he one presened here wihou habi formaion. 8 Recenly, Cochrane, Longsaff and Sana Clara (2004 managed o solve in closed form he case where n = 2, dividends are log-normally disribued, and agens are endowed wih log uiliy. Their modeling device seems hard o generalize for n > 2 and differen preferences. 5

7 o beer undersand he resricions ha have o hold in a general equilibrium seing and he naure of our assumpions below, i is insrucive o review he naure of he difficuly, as also explained in Sanos and Veronesi (2005. Le D = ( D 1,..., D n be he vecor of dividends, and assume for insance ha dividends are given by dd i D i = µ i D (D d + ν idb (4 for some drifs µ i D (D, and where ν i is a n 1 consan vecor, and db is a n 1 vecor of Brownian moions. From equaion (3 and Io s lemma, we find ha he process for aggregae consumpion is hen dc = µ c (s d + σ c (s db (5 C where s = ( s 1,..., s n = ( D 1 /C,,..., D n /C are shares of consumpion produced by dividends, and µ c (s = s i µ i D and σ c (s = i=1 s i ν i (6 The main difficuly in obaining racable and inerpreable formulas for asse prices lies in he dependence of he drif and he volailiy of he consumpion process on he shares s = ( s 1,..., s n. Sill, analyical formulas for asse prices can be obained by making economically plausible assumpions on he join processes of consumpion C and shares s, as advanced in MSV and Sanos and Veronesi (2005. Here we follow Sanos and Veronesi (2005 and make wo assumpions: Assumpion 1: The aggregae consumpion is given by i=1 where dc C = µ c (s d + σ c db µ c (s = µ c + µ c,1 (s and µ c,1 (s = s θ CF. (7 Above, θ CF = ( θcf 1,...,, θn CF and σc = (σ c, 0,..., 0. The specificaion of θcf i is explained below. Assumpion 2: For each i, he share s i follows he mean revering process ds i = φ ( s i s i d + s i σ i (s db (8 where σ i (s = ν i s j ν j (9 6

8 The cash-flow model (8 imposes a srucure on he relaive size of firms, where size is measured as he fracion of oal oupu produced by a given firm. In paricular, i imposes he economically plausible assumpion ha no firm will ake over he economy, as s i > 0 for all i. Finally, he volailiy σ i (s in (9 ensures ha n i=1 si = 1 for all. I is worh noing ha alhough he form of he volailiy σ i (s in (9 seems an ad-hoc formulaion, i acually sems from he model (4 - (5, as i is possible o verify by Io s lemma. II.C Cash-flow risk Given Assumpions 1 and 2, an applicaion of Io s Lemma shows ha dividends, D i = s i C, evolve according o he following process: dd i D i = µ i D,d + σ i D (s db (10 where µ i D, = ( s µ c + θcf i i + φ s i 1 (11 σd i (s = σ c + σ i (s (12 In hese formulas, θcf i = ν i σ c [ ] Firs, noe ha he expeced dividend growh µ i D, = E dd i depends on he relaive D i share s i /s i. When his quaniy is low, he asse s relaive conribuion o oal consumpion is below is long erm average and he asse has a higher expeced dividend growh. MSV es his predicion in a se of indusry porfolios and find srong suppor for i. In addiion, noe ha he long erm dividend growh of his asse is given by µ c, he uncondiional expeced reurn of consumpion growh, as well as a parameer θcf i, which is asse specific and i depends on he correlaion of he sock shares wih consumpion growh. Second, in his paper and because of he assumpion of habi formaion preferences, he sochasic discoun facor is only driven by shocks o consumpion growh. Thus, cash-flow risk is measured by he covariance of dividends wih consumpion growh, given by ( dd σcf, i i Cov D i, dc = σ c σ c + θcf i s θ CF (13 C The quaniy σcf, i, he condiional cash-flow risk of asse i, will play a prominen role in his paper. The erm θcf i s θ CF is paramerically indeerminae, ha is, adding a consan 7

9 o all θcf i leaves his erm unaffeced, as n i=1 si = 1. For his reason, we also impose he idenifiabiliy resricion s j θ j CF = 0. (14 Thus, we obain ha he uncondiional covariance beween asse i s cash-flow growh and consumpion growh is given by σ i CF = E [ ( σcf, i ] dd i = E [Cov, dc ] = σ c σ c + θcf i. (15 C In oher words, he parameer θcf i regulaes he relaive cash-flow risk of individual asses, as i is linear in he uncondiional covariance of dividend growh wih consumpion growh σ i CF. I is useful o emphasize ha he benchmark level of risk of an asse is he riskiness of aggregae consumpion. An asse is deemed risky (safe if is cash flows are more (less risky han aggregae consumpion. This is a general equilibrium resricion, as by definiion, he variance of consumpion growh mus be a weighed average of is covariances wih individual dividend growh. In wha follows we use he expression cash-flow risk when we refer o eiher σ i CF or θi CF as here is a one o one mapping beween one and he oher. As a final remark on cash-flows, noe ha he model is inernally consisen: If we apply he general equilibrium resricion on he drif of he consumpion process, (6, o he dividend process (10 [ ] dc E = C D i s i µ i D, = µ c + s θ CF, i=1 which equals (7 in Assumpion 1. Noe hen ha consumpion growh is no i.i.d. bu raher has some predicable componens which are linked o variaion in he vecor of shares, s. Sill, as we show below here is very lile predicabiliy in pracice as he parameers θ i CF small. 9 II.D Habi Dynamics are very As advanced by Campbell and Cochrane (1999, he fundamenal sae variable driving he aiudes owards risk in he habi model (1 (2 is he surplus consumpion raio, S = (C X C 1. In order o obain closed form soluions for prices when here are muliple 9 Briefly, we show in simulaions below ha expeced consumpion growh flucuaes beween a maximum of 2.22% and a minimum of 1.87%, a very mild variaion compared o he 1.5% sandard deviaion of consumpion growh ha we assume. Indeed, consisenly wih he empirical evidence, predicabiliy regressions in our simulaed daa produce a negligible level of predicabiliy of consumpion growh. 8

10 securiies, MSV model he inverse surplus consumpion raio Y = S 1 as a mean revering process. Unforunaely, heir modelling device canno be applied when γ > 1. In addiion, hey only obained approximae formulas for he case where θcf i 0. For hese reasons we op here for a differen sraegy and model direcly he process ( γ C G = = S γ (16 C X To model he dynamics of G sar by noicing ha one imporan difference wih respec o he models of Campbell and Cochrane (1999 and MSV is he fac ha in our model consumpion has some predicable componens. Thus i is imporan o specify he process for G in a way ha is consisen wih a ime-varying expeced consumpion growh. To have guidance on he ype of process for G, i is insrucive o consider he implicaions for G under he sandard assumpion ha X is an exponenially weighed average of pas consumpion levels, as in Consaninides (1990 and Deemple and Zapaero (1991, ha is, X = λ e λ( τ C τ dτ Io s Lemma immediaely shows ha dx = λ (C X d. Thus, an applicaion of Io s Lemma o (16 yields he process dg = [µ G (G σ G (G µ c,1 (s ] d σ G (G σ c db 1, (17 where µ G (G and σ G (G > 0 are complicaed funcions of G, provided in equaions (31 and (32 in he Appendix. Equaion (17 provides a srong inuiion on how he ime varying componen of he drif rae of consumpion µ c,1 (s should ener he process for G = S γ. Namely, a higher expeced consumpion growh µ c,1 (s implies a lower drif rae of G. The inuiion is ha an increase in he expeced growh rae of consumpion leads o a predicion of higher fuure consumpion compared o curren habi X and hus a higher surplus consumpion raio S in he fuure. Thus, given (16, his implies a lower expeced G. In wha follows, as in MSV and Campbell and Cochrane, we simplify (17 o obain a more manageable process, bu we reain he inuiion ha µ c,1 (s eners he drif rae of G, in he same form as specified in (17 and assume he following drif and diffusion processes: µ G (G = k ( G G Noice hen ha he drif of G has wo componens o i. and σ G (G = α (G λ. (18 The firs one is a mean reversion componen and capures he basic idea of habi persisence models, namely ha 9

11 he habi X evenually caches up wih C. The second componen links he drif rae of G o he ime varying componen of he drif rae of consumpion growh. Once again, in order o reain he habi formaion formulaion, i is imporan o assume ha he coefficien ha muliplies µ c,1 (s equals he diffusion erm in he process iself. As for he diffusion componen, as in MSV, λ 1 bounds G from below a λ and α > 0 ransmis he innovaions in consumpion growh, db 1, o he convexiy of he uiliy funcion. These assumpions allow us o obain closed form formulas for asse prices. Noe ha he habi model of MSV is a special case of (17 and (18: In fac, MSV assume ha γ = 1, and ha consumpion growh is i.i.d., an assumpion ha obains here by seing µ c,1 (s = 0. III. EQUILIBRIUM ASSET PRICES AND RETURNS In his secion we characerize he prices and reurns associaed wih he economy described in he previous secion. Our sraegy o do his is sandard. Given (2, he sochasic discoun facor is given by m = e ρ (C X γ = e ρ C γ G. Then we can use Io s Lemma, ogeher wih our assumpions on he dynamics of C and G = S γ o show ha dm m = r f d + σ mdb, he firs, and only non-zero, enry in he diffusion componen vecor, σ m, is given by σ 1 m = [γ + α (1 λs γ ] σ c. (19 Then we exploi our assumpions on he dynamics of C, G = S γ ] [ = E P i = E [ ( mτ m D i τ dτ ( mτ m and s i o solve for ] s i τ C τ dτ in closed form. We hen use our expressions for prices o compue reurns, (20 dr i = dp i + D i d P and calculae he expeced excess reurns ( [ ] E dr i dm = cov, dr i m rd = σ mσ i R, (21 10

12 where σr i is he diffusion componen associaed wih he reurns of asse i. Our purpose is o obain closed form expressions for (20 and (21 and relae hem o he parameers in our model. III.A The oal wealh porfolio I is useful o sar characerizing some basic properies of he oal wealh porfolio as he inuiion for some of hese resuls becomes useful laer. Proposiion 1: The price-consumpion raio, he expeced excess reurn and diffusion erms of he oal wealh porfolio are, respecively: P T W C = α T W 0 (s + α T W [ ] E dr T W σ T W where he funcions α T W 0 (s,α T W 1 (s, f T W 1 (s and 1 (s S γ (22 = (γ + α (1 λs γ S γ α (1 λsγ f1 T W (s + S γ σc 2 + wj T W σ j CF, (23 R, = Sγ α (1 λsγ f1 T W (s + S γ σ c + wj T W σ j D (s, (24 { } are given in Appendix. w T W j The resuls are similar o he ones found by Campbell and Cochrane (1999 and MSV, and we refer o hose papers for more deail. Briefly, he price-consumpion raio of he oal wealh porfolio is increasing in he surplus consumpion raio S. This is inuiive: a high surplus consumpion raio implies a low local curvaure of he uiliy funcion, a less risk averse aiude by par of he represenaive agen, and his in urn ranslaes ino higher price-dividend raios. Differenly from Campbell and Cochrane (1999 and MSV, he oal wealh porfolio depends also on he enire vecor of shares s. The reason is ha in our seing, he general equilibrium resricion (5 generaes a mild predicabiliy in consumpion growh (see Assumpion 1. The funcions α T W 0 (s and α T W 1 (s are ypically decreasing in expeced consumpion growh, because in our se up he elasiciy of ineremporal subsiuion is less han one. Thus, his componen implies ha an increase in µ c (s resul in lower prices To review he economic reasoning, a low elasiciy of ineremporal subsiuion implies a desire for consumpion smoohing. Thus, an increase in expeced consumpion growh yields a higher desire of curren consumpion, and hus lower savings. The consumer hen sell socks and bonds, resuling in a decrease of he P/C raio of he oal wealh porfolio. 11

13 Turning o he expeced excess reurns, he firs erm in parenhesis reflecs he curvaure of he uiliy funcion of he represenaive agen, and hus he degree of risk aversion: High curvaure parameer γ or low surplus S imply high expeced reurns. The firs erm of he expression in brackes is insead linked o discoun effecs: As shown in he pricing funcion, changes in S induce a volailiy of sock reurns which is perfecly correlaed wih he sochasic discoun facor, and hus i is priced. MSV discuss his effec more horoughly. The novel erm is he second one in he bracke, which is he premium invesors require because of changes in expeced consumpion growh. This second erm is ypically negaive. The reason is ha our modeling device induces a mild posiive correlaion beween shocks o consumpion growh and shocks o expeced consumpion growh. Thus, since as explained earlier, negaive shocks o consumpion growh are correlaed wih posiive shocks o prices, his componen carries a negaive premium. III.B. Prices and reurns for individual securiies Proposiion 2: The price of asse i is given by P i ( ( = α0 i + α1s i γ s i + αi 2 (s + α3 i (s S γ s i D i s i s i (25 where α0 i, αi 1 are posiive consans and αi 2 (s and α3 i (s are posiive funcions of he share vecor s given in appendix. Proposiion 2 shows ha he price-dividend raios of asse i is increasing in he surplus consumpion raio S. As i was rue for he oal wealh porfolio, a high S implies a lower risk aversion of he represenaive agen, and hus higher prices of asses. In addiion, however, he price-dividend raio is increasing in he relaive share, s i /s i, which as shown in expression (11, deermines he ime varying componen of expeced dividend growh. A high expeced dividend growh resuls naurally in a higher price-dividend raio. The las erm shows ha shocks o expeced dividend growh have a differen effec depending on he level of he surplus consumpion raio: The higher he surplus consumpion raio he sronger he impac of expeced dividend growh shocks on he price-dividend raio. Finally, as i was rue for he oal wealh porfolio, he price of each individual asse also depends on funcions of he vecors of shares α2 i (s and α3 i (s. As explained earlier, hese funcions are ypically decreasing in expeced consumpion growh because he elasiciy of ineremporal subsiuion is less han one in our seing. Thus, an increase in he expeced consumpion growh decreases price-dividend raios. 12

14 Proposiion 3: (a The diffusion erm of he reurn process for asse i is given by σr, i = Sγ ( α (1 λsγ σ s i c + 1 ( + η i, s + S γ 1 + f2 i (S, s s i i σd i (s + ηjσ i j D (s (26 j i µ DISC f i 1 s i (b The expeced excess reurn of asse i is given by s i E [ dr i ] = µ DISC i, + µ CF i, where i, = (γ + α (1 λs γ αsγ ( (1 λsγ σ f1 i s i, s s i + S γ c 2 (27 µ CF i, = (γ + α (1 λs γ 1 ( + η i 1 + f2 i (S, s s i i σcf, i + ηjσ i j CF, (28 j i s i wih f1 i ( s i /s i α i, s = 0 + αi 2 (s ( s i /s i α1 i + αi 3 (s ( s i /s i > 0 and f2 i (S, s = αi 2 (s + α3 i (s S γ α0 i + αi 1 Sγ > 0, and η i j are given in Appendix. Proposiion 3 shows ha he expeced excess reurns he effecs are divided in wo. These wo erms correspond o he wo sources of shocks o reurns ha are shown in he expression for he diffusion componen of reurns, (26. The firs correspond o shocks in he discoun facor and he second o cash-flow shocks, boh is own, he second erm in expression (26, and shocks o he cash-flows in he res of he asses in he economy. III.B.1 Discoun risk effecs The source of his componen of he risk premium, µ DISC i,, is he variaion of he aggregae discoun proxied by S γ. To inerpre furher his erm i is useful o noice firs ha P i /P i S γ /Sγ S γ = ( f 1 s i /s i, s + S γ. (29 Thus he volailiy of an asse s reurn is linked o he elasiciy of prices o shocks in he variable driving he aggregae discoun, which is S γ. The variance of hese shocks is linked o α (1 λs γ σ c, 13

15 which is he diffusion componen of ds γ /Sγ, he inverse of our sae variable G, as i follows from a basic applicaion of Io s Lemma o (17. Clearly, only he componen of hese shocks ha covaries wih he shocks o he sochasic discoun facor is priced which, given (19 is [γ + α (1 λs γ ] α (1 λsγ σ2 c. (30 The componen of he asse s premium ha is linked o discoun effecs is he produc of (29 and (30. Cross-secional variaion in he discoun effecs can only be driven by differences in he price elasiciy (29, which is in urn driven by he behavior of he funcion f 1 ( s i /s i, s. We have been unable o obain a general characerizaion of his funcion, bu for parameer values ha are empirically relevan we find ha ( f 1 s i /s i, s ( s i /s i < 0, and hus asses wih a higher expeced dividend growh, as measured by he relaive share s i /s i, display sronger discoun effecs. The inuiion is sraighforward: socks wih a high expeced dividend growh pay he bulk of is proceeds far in he fuure. Thus, minor variaions in he aggregae discoun rae hrough he risk aversion of he represenaive invesor resul in large percenage variaions of he price of he asse, as also shown in he firs erm of he diffusion funcion σr, i in equaion (26. This variaion is naurally priced and hus he higher required premia associaed wih asses wih large relaive shares. III.B.2 Cash-flow risk effecs The source of premia relaed o cash-flow shocks, µ CF i,, has wo componens o i. The firs is relaed o shocks in he asse s dividends, which is he second erm in expression (26, and he second is relaed o shocks in he dividends of he res of he asses in he economy, which, as shown in (25, affec he price of asse i as well. The logic for he sources of he premia linked o cash-flow shocks is he same as in he discoun effecs case. Firs i can be easily shown ha he elasiciy of he price wih respec o shocks o is own dividends is, P i /P i D i /Di 1 = 1 + f2 i (S, s ( s i s i + η i i. The diffusion erm of he dividend process of asse i is σ i D (s, and recall ha is condiional covariance wih consumpion growh is denoed by σcf, i. The firs componen of µcf i, is hen he componen of he dividend shocks ha covaries wih shocks o he sochasic discoun 14

16 facor muliplied by he effec ha hese shocks have on he price of asse i, as measured by he elasiciy. A similar logic applies o he second erm in µ CF i,. Indeed i can be shown ha P i /P i D j /Dj = η i j for j i. As before his componen of he premium resuls from he produc of his (cross elasiciy and he priced componen of he shock o asse j s dividends, σ j CF,. How does he curren level expeced dividend growh, as measured by s i /s i, affec he cash-flow risk componen of expeced sock reurns? Given he condiional covariance of he dividend of asse i wih aggregae consumpion, σcf, i, he firs erm of (28 is unambiguous: Since f2 i (S, s > 0, if he asse is risky, ha is, if σcf, i > 0, hen a high expeced dividend growh ranslaes in a lower premium semming from curren dividend volailiy. The inuiion is also clear: a sock ha pays more in he fuure han oday has a relaively low dividend compared o he fuure. Thus, he risk embedded in curren dividends, σcf, i, has a relaively low impac on he oal risk of sock. In he limi, if he socks does no pay any dividend oday, i canno have any cash-flow risk, as here is zero curren covariance of dividends wih consumpion. If insead he asse s dividends covary negaively wih consumpion growh (σ i CF, < 0, hen a high expeced dividend growh increases he risk premium. The argumen, of course, is he converse of he previous one. The effec ha he curren expeced dividend growh of asse i has on he second erm of he cash-flow risk componen of sock reurn reurn (28 is more difficul o inerpre. To quanify hese effecs, he op panel of Figure 1 plos he quaniy µ CF i, as a funcion of he ] uncondiional cash-flow risk σ i CF [σ = E CF, i a he seady sae, ha is, for he case where S = S and s = s. As i can be seen, he cash-flow componen of expeced reurn is increasing in σ i CF. Noe however, ha here is a negaive bias in his componen of expeced excess reurn. Indeed he case σ i CF = 0 sill implies a negaive expeced excess reurn semming from cash-flow risk effecs. This is due o he second componen in (28, which is relaed o he ime variaion in he aggregae expeced consumpion growh. As we discussed in he case of he oal wealh porfolio, his componen carries ypically a negaive risk premium. Finally, he boom panel of Figure 1 plos µ CF i, as a funcion of σ i CF for he case where S = S bu for a random draw of shares s. Alhough an increasing paern in σcf i can be easily seen, crosssecional differences in s i /s i may make he componen µ CF i, of an asse wih high uncondiional cash-flow risk σ i CF emporarily lower han ha of an asse wih lower cash-flow risk σi CF. 15

17 III.C The value premium In order o gauge he source of he value premium in our model i is convenien o urn o Figure 2. Panels A, B, and C plo µ DISC i,, µ CF i,, and he oal E [ ] dr i respecively agains he relaive share s i /s i for various levels of he asse s uncondiional cash-flow risk σ i CF, which correspond o differen values of θ i CF (see expression (15. In all cases, he level of surplus S is se o is seady sae value S. The parameers used are hose of he calibraion exercise discussed in deail in he nex secion. Sar wih Panel A. As discussed in Secion III.B.2, he discoun risk componen of expeced reurn is increasing in he relaive share s i /s i, ha is, wih expeced dividend growh. The reason is ha asses wih high relaive shares are more sensiive o shocks in he sochasic discoun facor. These shocks are naurally priced and hus he higher required premia of asses wih high relaive shares. In addiion, he discoun risk componen of expeced reurns does depend as well on he asse s uncondiional cash-flow risk σ i CF : Socks wih higher cash-flow risk σ i CF have a larger discoun risk componen in expeced reurns. The inuiion is ha socks wih a higher σ i CF are riskier and as a consequence have lower prices. I follows ha changes in he sochasic discoun facor have a larger impac, in percenages, on he prices of asses wih higher levels of cash-flow risk. Panel B of Figure 2 plos he cash-flow risk componen of expeced reurns which, as discussed in Secion III.B.2, is decreasing in expeced dividend growh for socks wih high cash-flow risk. Finally, Panel C repors he oal expeced reurn for each asse ha is obained by adding o he discoun risk componen he cash-flow risk componen of sock reurns. III.C.1 Discoun risk effecs and he growh premium Wha does his decomposiion of expeced excess reurns imply for he value premium? Noice firs ha, given expression (25, soring asses according o heir price-dividend raio is akin o soring hem on boh cash-flow risk, σ i CF, and expeced dividend growh si /s i. In paricular, value socks (asses wih low P/D raios are, on average, associaed wih high σ i CF and low expeced dividend growh si /s i. Consider now he case where cross-secional differences in cash-flow risk are small (e.g. θcf i 0 for all i. Then, σi CF are roughly he same across all asses and hus he soring procedure simply picks differences in expeced dividend growh as measured by s i /s i. In his case, he oal expeced excess reurn are as in Panel A as discoun effecs dominae. Since low price-dividend raio socks are hose corresponding o low relaive shares s i /s i, value socks are found on he lef-hand side of he panel and hus have low expeced excess reurns. Similarly, high price-dividend raio socks 16

18 are hose wih high s i /s i and hus growh socks are found on he righ-hand side of he panel and have high expeced excess reurns. To summarize, if cross-secional differences in cashflow risk are small, hen growh socks have higher expeced excess reurns han value socks. Tha is, a growh premium obains. III.C.2 Cash-flow risk effecs I follows from he discussion above ha if a value premium is o obain here mus be subsanial cross-secional differences in cash-flow risk. To see his, consider now Panel C, which repors he oal expeced reurn when boh discoun effecs (Panel A and subsanial cash-flow effecs (Panel B are presen. Value socks (asses wih low P/D raio have, on average, high risk, ha is, high σcf i, and low expeced dividend growh, ha is, low si /s i. This combinaion corresponds o he area around he op-lef corner of he plo, ha is, o high expeced excess reurn. Conversely, growh socks (asses wih high P/D raios mus have a combinaion of low σcf i and high si /s i. This combinaion can be found on he boom-righ corner of he plo. As i can hen be seen hen value socks will command a high premium and growh socks a low (and even negaive premium. To summarize hen if cross-secional differences in cash-flow risk are large, hen value socks have higher expeced excess reurns han growh socks. Tha is, a value premium obains. III.C.3 Cash-flow risk effecs and he dynamics of he value premium Campbell and Cochrane s (1999 imporan conribuion is o show how a srong variaion in risk preferences is able o generae he main ime series properies of he aggregae marke porfolio, including a high equiy risk premium, large volailiy of reurns, a low and consan ineres rae, a sizable Sharpe raio, and, imporanly, a srong predicabiliy in aggregae excess reurn. 11 We have shown ha in such an environmen, if cross-secional differences are mainly driven by cross-secional differences in expeced dividend growh, hen a growh premium obains raher han a value premium. Given ha wha is empirically observed is he laer and no he former, i mus be he case ha cross-secional differences in cash-flow risk mus be he main deerminan of he cross-secion of sock reurns. Clearly, he size of hese cash-flow risk effecs can only be assessed in a model where he srong discoun risk effecs needed o generae he ime series properies of he marke porfolio are presen, oherwise one would underesimae he srengh of he cash-flow risk effecs ha are in urn needed 11 More generally, Melino and Yang (2003 have shown in he conex of a simple and pedagogical example ha a very sensiive sochasic discoun facor is needed o mach he basic properies of he marke porfolio. In paricular wha is required is a high variaion in he condiional variance of he sochasic discoun facor. 17

19 o obain he value premium. The presence of discoun risk effecs which are associaed wih he ime series variaion in risk preferences has addiional implicaions in wha concerns he dynamics of he value premium. Essenially, discoun risk effecs inerac wih he cross-secional dispersion in cashflow risk o induce flucuaions in he value premium, as shown in Figure 3. This figure plos he expeced excess reurns of hree asses agains he surplus consumpion raio, S. The doed line shows he expeced excess reurn for he marke porfolio; he solid line corresponds o he expeced excess reurn on a represenaive value sock wih high cash-flow risk and low expeced dividend growh; finally he dash line corresponds o he premium of a represenaive growh sock wih low cash-flow risk and high expeced dividend growh. As i can be seen, when he surplus consumpion raio is low (high, he value premium is high (low. This is inuiive: In our framework asses wih a high value of θ i CF are paricularly riskier when he represenaive agen s is highly risk averse which occurs whenever adverse consumpion growh shocks depress he surplus consumpion raio, increasing in urn he marke premium and is dividend yield. Thus in our model he value premium has a srong predicable componen being high (low when he marke premium is high (low. IV. THE VALUE PREMIUM AND ITS DYNAMICS In his secion we conduc a simulaion sudy o evaluae he exen o which he model can mach he momens of he reurn daa ha have become sandard in he lieraure boh in he ime series and he cross-secion. These momens can be found in Table I. The daa se is sandard and i is very briefly described in he Noes o Table I. Panel A shows mean and sandard deviaion for he reurns on he marke porfolio and he risk free rae. Panel B shows he predicabiliy regressions of Fama and French (1988 and Campbell and Shiller (1988, for wo differen sample periods, which are mean o emphasize ha long horizon reurn forecasabiliy is sensiive o he paricular period under consideraion. Panels A and B are he sandard concern of he equiy premium lieraure. Panel C shows he value premium and is corresponding puzzle, he failure of he CAPM o generae he large cross-secional dispersion in average reurns across book-o-marke sored porfolios. Panel C is he sandard concern of he cross-secional lieraure. We leave he sudy of he CAPM s failure o explain he value premium for Secion V.A. The abiliy of a model o mach he sylized paerns in Panels A and B is essenially relaed o he properies of he sochasic discoun facor ha he model generaes. As i is 18

20 well known, habi persisence models à la Campbell and Cochrane, such as he presen one, are relaively successful in reproducing hese paerns in he ime series of aggregae reurns. Panel C insead is relaed o boh discoun and cash-flow effecs and i is he focus of our invesigaion bu wihou loosing sigh of he model s abiliy o say close o he facs in Panel A and B. The presence of discoun risk effecs also generaes implicaions for he ime series behavior of he value premium and we sudy hese effecs in Secion IV.C. IV.A Deails of he simulaion We simulae he model presened in Secion II.B wih 10,000 years of quarerly daa for 200 firms. We sor hese asses ino en porfolios according o heir price-dividend raio in an effor o mimic he sandard procedure used in he cross-secional lieraure and focus our analysis on hese en porfolios. Table II conains he parameer values ha are going o be used hroughou. These values were chosen o generae momens for our simulaed economy ha are close o heir empirical counerpars in Table I. Panel A of Table II conains our choice of parameers for he consumpion process and preferences. Average consumpion growh, µ c, is se a 2% and he sandard deviaion is se a 1.5%. This laer value should be measured agains a sandard deviaion of consumpion growh for he poswar sample of 1.22% and he one for he longer sample saring in 1889, which is 3.32%. 12 As for he preference parameers, our choice of γ is beween he values used by MSV, γ = 1, and he value used by Campbell and Cochrane (1999, γ = 2. Alhough he process ha is modeled is G = S γ, i is more inuiive and clear o hink abou our choices in erms of he local curvaure of he uiliy funcion, γs 1, as i is he economically relevan magniude for he effecs of ineres. Our choices imply a seady sae value of he local curvaure of he uiliy funcion of γs 1 = 48, higher han he already high value of Campbell and Cochrane (1999 which is 35. The minimum value of his local curvaure is Finally he parameer k and α are similar o he values chosen by MSV. As for he share process, we assume ha all of he 200 simulaed asses have he same seady sae conribuion o overall consumpion, s i = 1/200 =.005. Also he speed of mean reversion is se a φ =.052, a value ha is only slighly lower han he one esimaed by MSV for he overall marke porfolio. The key parameer of ineres in our model is he one ha conrols differences in cash-flow risk, θcf i. Our general equilibrium seing requires ha his parameer is symmerically disribued around zero, as he aggregae marke mus have a 12 See Campbell and Cochrane (1999 Table 2. 19

21 variance equal o he one of consumpion, ha is he cash-flow risk parameers mus be θ i CF [ θ CF, θ CF ], where θ CF > 0, so ha for any asse wih cash-flow risk θ i CF here is a mirror asse wih cash-flow risk equal o θcf i. Throughou, and wih some abuse of erminology, we refer o θ CF as he cash-flow risk parameer bu he reader should keep in mind ha i is he suppor of he cash-flow risk parameers of individual asses. In wha follows, we sar discussing a baseline case wih θ CF = for i generaes quaniaively plausible implicaions for he cross-secion of sock reurns. We invesigae his case in deail and hen, in Secion VI, we sudy he behavior of he model under differen assumpions on he size of he parameer θ CF. We also pospone a discussion of he size of he cash-flow risk effecs unil ha secion. IV.B The ime series properies of he marke and he value premium Table III is he analog o Table I bu in simulaed daa. Panel A shows he implicaions of our model for he aggregae daa. The model generaes a sizable, if slighly low, equiy premium and volailiy of sock reurns, and he risk free rae momens are reasonable. Panel B of Table III shows he predicabiliy regressions for all he sandard horizons. As already menioned he model does well in his dimension: The coefficiens all have posiive signs and increase wih he forecasing horizon as do he saisics. The R 2 s are relaively lower han heir empirical counerpars bu no far off he mark for he case of he sample. These resuls simply reproduce he good performance of he model in Campbell and Cochrane (1999 and MSV in wha refers o he marke porfolio. Panel C of Table III conains he average excess reurns for he en porfolios sored on P/D. The value premium obains nicely in our seup. Indeed he value premium is a healhy 5.88%, only slighly above he empirically observed one of 5.50%. Noice hough ha he average excess reurns for each porfolio are below heir empirical counerpars. The reason is ha as menioned he model misses he equiy premium by abou 3%. This low premium can be seen as well in he Sharpe raio bu, imporanly, hey decrease wih he price-dividend raio, an imporan feaure of he daa (see Table I. The line denoed Avge ( θ i CF 100 repors he average cash-flow risk parameer for each of he decile porfolios. As he inuiion developed in Secion III.C suggesed, he soring procedure picks cross-secional variaion in he cash-flow risk parameer, θ i CF : Socks in he value porfolio, porfolio 10, have, on average, a high cash-flow risk parameer whereas he 20

22 opposie is rue for he growh porfolio, porfolio 1. In our framework, and in line wih much of he recen empirical research on his issue, 13 value socks are indeed riskier in he cash-flow sense and he srengh of his effec is enough o undo he naural discoun riskiness of growh socks. Indeed, as shown also in Secion III.C and in paricular in he op panel of Figure 2, if discoun risk effecs dominae a growh premium obains. Subsanial discoun risk effecs are needed o mach he ime series properies of he marke porfolio hough and we show in he nex secion ha hese effecs inerac in urn wih he cross-secional dispersion of cash-flow risk o generae ineresing dynamics for he value premium. IV.C The dynamics of he value premium As i is well known, habi persisence models induce flucuaions in he represenaive agen s aiudes owards risk, which in urn ranslae ino predicable and highly volaile sock reurns. In his secion we show ha hese flucuaions also inerac wih he crosssecional dispersion in cash-flow risk o induce in urn a predicable and volaile value premium. These flucuaions in he value premium are argued as a primary reason for he failure of he uncondiional CAPM, a opic o which we reurn below. To ascerain he ime series variaion of he value premium Table IV Panel A shows he average excess reurn of he firs and enh decile porfolio as a funcion of wheher he markeo-book raio of he marke porfolio is above or below a cerain percenile, denoed by c. For insance, he firs line shows ha he average excess rae of reurn of he firs decile (growh porfolio is 13.18% if he marke-o-book of he marke porfolio is below he 15h percenile of is empirical disribuion and ha of he enh decile (value porfolio is 23.57%. The value premium is hen 10.38%. Insead when he marke-o-book is above he 15h percenile he firs decile porfolio has an average excess reurn of 5.73% and he enh porfolio has one of 10.35% for a oal value premium of 4.62%, which is considerably lower han he previous one. This paern is consisen independenly of he percenile ha is chosen as a cu-off poin. The value premium is higher whenever he marke-o-book of he marke porfolio is low. Noice ha hese are also periods where he average excess reurn of he marke is high, as shown in he columns headed by R M. How does he model perform in his dimension? Panel B of Table IV repors he same calculaions as in Panel A bu wih simulaed daa. The only difference is ha, naurally, insead of using he marke-o-book we use he price-dividend raio of he marke porfolio o 13 See Bansal, Dimar, and Lundblad (2005, Campbell and Vuoleenaho (2005, Parker and Julliard (2005, and Hansen Heaon and Li (

23 idenify he sae. The paern is indeed very similar wih he only excepion of he level of he premia which is, as already discussed, lower han in he daa. The value premium is higher when he price-dividend raio of he marke porfolio is low han when i is high. In summary hen, he discoun risk effecs needed o replicae he ime series properies of he marke porfolio inerac wih he cross-secional dispersion in cash-flow risk o generae variaion in he value premium. Value socks are paricularly risky during bad imes, periods when he aggregae marke premium and is dividend yield are high relaive o heir uncondiional mean, an effec ha is presen boh in he daa and he model. V. THE CAPM AND OTHER ASSET PRICING MODELS As shown in he previous secion he model maches well he basic momens of reurn daa boh in he ime series and he cross-secion. Whereas several general equilibrium models have been pu forh o address he ime series properies of he marke porfolio his has been much less he case when i comes o he cross-secion of sock reurns. 14 Sill, several empirically based asse pricing models have been proposed o explain he paerns in he crosssecion. To wha exen is our model consisen wih hese empirically based asse pricing models? How can our model illuminae he apparen success (and failure of hese models? In his secion we address hese quesions. We focus our analysis on hree imporan models: The CAPM of Sharpe (1964 and Linner (1965, he Fama and French (1993 model and he condiional asse pricing models proposed of lae of which Leau and Ludvigson (2001 is he foremos example. I is imporan o emphasize ha in our framework, all hese models are misspecified bu as we show below, wih he excepion of he CAPM hey work raher well as hey capure complemenary aspecs of he daa. V.A The CAPM V.A.1 The CAPM and he value premium puzzle Any asse pricing model ha generaes he value premium has also o be consisen wih he value premium puzzle, he inabiliy of he CAPM o price marke value sored porfolios. 15 The value premium puzzle can be seen in he las line of Table I Panel C (CAPM 14 See Gomes, Kogan and Zhang (2003 for a noable excepion. 15 The inabiliy of he CAPM o explain he cross secion of average reurns is pronounced in he poswar sample used in his paper. Recenly hough Ang and Chen (2005 and Fama and French (2005 show ha he behavior of he CAPM in he earlier sample covering is much beer. Sill Daniel and Timan (2005, Table 3 and Fama and French (2005 perform riple sors, on ME, BE/ME and (preformaion marke bea o find variaion in average reurns unrelaed o bea hus rejecing he CAPM. 22